An adverse gradient grows the wake, not the log law

The wake is a tenth of the velocity and a third of the displacement put Coles’s wake on top of the law of the wall and found it carrying a third of a flat-plate layer’s displacement and deciding its friction law. The flat plate’s wake has one strength, about 0.55. That essay ended on the observation that a layer in a pressure gradient has another, and that the wake strength is the number in which a layer remembers what its outer flow has done to it. This essay follows the layer up a rising pressure.

The question matters because rising pressure is where boundary layers fail. How much uphill a layer can take asked it for a laminar layer, which separates after a modest pressure rise. A turbulent layer can climb much more, and every diffuser, every wing’s upper surface behind its suction peak and every ship’s stern depends on how much more.

Layers that stay alike #

A layer climbing an arbitrary pressure rise changes shape continuously, and its profile at any station depends on its whole history. Clauser in 1954 found a class of layers that do not: in a pressure gradient held in a fixed proportion to the layer’s own friction, the profile stays similar to itself as the layer grows. The proportion is

$$\beta = \frac{\delta^*}{\tau_w}\,\frac{dp}{dx},$$

the pressure rise over one displacement thickness against the wall’s shear. A flat plate has $\beta = 0$. Clauser built two layers in a wind tunnel with $\beta$ near 2 and near 8, adjusting the tunnel’s flexible wall until the outer profile stopped changing shape, and showed each has its own velocity-defect law — its own wake.

Equilibrium layers are the turbulent counterpart of the laminar Falkner–Skan family: the cases with no history, in which one number replaces the past. What connects $\beta$ to the wake strength is measured, not derived. The fit used here is Das’s, $\Pi = 0.8(\beta + 0.5)^{3/4}$, which gives 0.48 at $\beta = 0$, close to the flat plate’s measured 0.55, and grows without limit with $\beta$.

The laminar analogue makes the comparison sharp. A laminar layer in a power-law stream is one of the Falkner–Skan family, and it separates at $m = -0.0904$: a stream decelerating as the inverse eleventh root of the distance is all a laminar layer can follow. The turbulent equilibrium layers below follow decelerations nearly three times as steep. The difference is entirely in how momentum reaches the wall — by viscosity alone in one case, by eddies the size of the layer in the other.

The profile hollows from the outside in #

The composite profile with the fitted wake shows what an adverse gradient does. At $\beta = 0$ the profile is the full turbulent one, rising steeply near the wall and flat across most of the layer. At $\beta = 2$ the wake strength is 1.59; at 10, 4.67; at 50, 15.2. The wall region keeps its law throughout — the profile in wall units near the wall is the same in every case — but the wake grows until it is most of the edge velocity, and the profile sags: most of the layer is moving slowly, above a thin wall region that is still being driven by the friction.

That is the opposite of the laminar picture, where a rising pressure attacks the fluid next to the wall first and the profile develops an inflection close to it. The turbulent layer’s slow fluid accumulates in the outer part, and the wall region, kept moving by the momentum the large eddies bring down to it, holds on.

Fuller in shape, weaker at the wall #

In numbers, at a momentum-thickness Reynolds number of $10^4$, the shape factor rises from 1.32 on a flat plate to 1.91 at $\beta = 10$ and 3.1 at $\beta = 100$, and the friction coefficient falls by 2.8 times at $\beta = 10$ and 27.5 times at $\beta = 100$. The friction falls steadily and never reaches zero: at any finite $\beta$ the wall region is still there and still being sheared.

That is the content of “an equilibrium layer approaches separation without arriving”. Separation, the vanishing of the wall shear, is the limit $\beta \to \infty$ — a gradient infinitely strong compared with a friction that has fallen to nothing — and a layer can be held arbitrarily close to it, at a friction as small as one likes, as long as the gradient is matched to the friction at every station. Stratford built such a layer in 1959, a turbulent layer on the point of separation all along its length, and it remains the steepest pressure rise a layer can be made to climb without separating.

Why the wall region holds on #

The friction falls and the wall region survives, and the mechanism is the large eddies. In a turbulent layer the momentum that keeps the fluid near the wall moving is not diffused down from the free stream by viscosity; it is carried down by the turbulent motions, fast fluid swept towards the wall and slow fluid ejected from it, and the average of that exchange is the Reynolds stress — the quantity the mean is not the flow is about. As the wake grows the outer fluid is slower, so each sweep brings down less momentum than it would on a flat plate, and the wall shear falls. But the sweeps continue as long as the layer is turbulent, and so does the supply. The wall region is starved, not cut off.

A laminar layer has no such supply. The only way momentum reaches its wall is down the velocity gradient by viscosity, a slow process that a rising pressure can outpace within a short distance, and then the fluid at the wall stops and turns back. That is the whole of the turbulent layer’s advantage in climbing a pressure rise, and it is why every device that must recover pressure from a fast stream — a diffuser, a wind tunnel’s return circuit, a jet engine’s intake duct, the back of a wing — is designed to keep its boundary layers turbulent.

Reading β off a wing #

The parameter is easy to estimate from quantities a wind tunnel measures. Behind the suction peak of a wing in a thirty-metre-a-second stream, a layer with a displacement thickness of five millimetres, a wall shear of one pascal and a pressure rising at two hundred pascals a metre has $\beta = 1$; the same layer near the trailing edge, thicker and with half the shear under a steeper rise, is at five or ten. So the range drawn here, from zero to a hundred, spans a wing’s upper surface from just behind its suction peak to the brink of trailing-edge separation, and the shape factors of 1.4 to 3 measured along such a surface are the equilibrium values at those $\beta$, roughly — roughly because a wing’s $\beta$ changes along it, and its layer is always catching up with the value it has just left. The drag a wing pays for having a boundary layer at all is set mostly by the momentum thickness this layer arrives at the trailing edge with, which is why the aft pressure distribution of a section is designed with this arithmetic in mind.

The Reynolds number hardly moves it #

The equilibrium values depend on the Reynolds number, but weakly. At $\beta = 5$ the shape factor is 1.80 at a momentum-thickness Reynolds number of 3,000, 1.68 at $10^4$, 1.60 at $3\cdot10^4$ and 1.53 at $10^5$, and the ratio of the flat plate’s friction to this layer’s falls from 1.98 to 1.69 over the same range. Both move as the logarithm of the Reynolds number moves, for the reason the flat plate’s shape factor does: the wake’s size in wall units is fixed by $\beta$, while the logarithm’s share of the edge velocity grows with the layer. A model tested at a tenth of full-scale Reynolds number shows a shape factor a few per cent higher and a friction ratio a few per cent larger than its full-size counterpart — a scale effect that, like the roughness a wall cannot feel, is real and predictable rather than a reason to distrust the test.

The steepest climb #

An equilibrium layer needs a particular outer flow, and the momentum integral says which. With a free stream $U_e \propto x^m$ and a momentum thickness growing in proportion to distance, the momentum integral and the definition of $\beta$ together give

$$m = -\frac{\beta}{H(1 + \beta) + 2\beta}.$$

At $\beta = 0$, $m = 0$: a flat plate. As $\beta$ grows $m$ becomes negative — a decelerating stream — but $H$ grows too, and the two compete. The result is the figure’s surprise: $m$ falls to a least value of $-0.249$ near $\beta = 4.8$ and then rises again. Layers nearer separation, at larger $\beta$, need a gentler deceleration, not a steeper one, because their larger shape factor means each unit of deceleration costs them more momentum thickness.

So there is a steepest equilibrium pressure rise: a free stream falling no faster than about the inverse fourth root of the distance. A diffuser whose wall is shaped so that its core velocity falls faster than that cannot hold an equilibrium layer on its walls; the layer must thicken faster than equilibrium allows and is on its way to separation. The value computed here, $-0.249$, is close to the $-0.23$ or so usually quoted for an equilibrium layer at the point of separation — the difference is the composite profile’s own approximation, and the existence of the extremum does not depend on it.

A diffuser held at the limit #

The steepest equilibrium rise translates directly into a diffuser’s length. With the core velocity falling as $x^{-1/4}$, the pressure recovered between two stations is $1 - (x_1/x_2)^{1/2}$ of the inlet’s dynamic pressure: half of it over a fourfold increase in distance from the virtual origin, three-quarters over sixteenfold. A diffuser shaped to keep its wall layers in equilibrium at the steepest rise therefore recovers pressure slowly and steadily, and a designer who wants the recovery faster must accept a layer that is not in equilibrium and is closer to separation than its local shape factor suggests. Stratford’s layer, held at zero friction throughout, recovers pressure along almost exactly this kind of law, and it was designed as the fastest possible recovery that does not separate. A duct that diffuses behind a turbine faces the same arithmetic: the exit area it can use is set by how fast its walls’ layers can be decelerated without letting go.

A layer that is all wake #

As the wake strength grows without limit, the wall law’s share of the profile shrinks to nothing and the profile tends to the pure wake, $u/U_e = \sin^2(\pi y/2\delta)$. Its thicknesses are exact: the displacement thickness is half the layer, the momentum thickness an eighth, and the shape factor exactly 4. The composite layer approaches it slowly: 2.4 at $\Pi = 10$, 3.99 at a thousand.

Measured turbulent separation happens at shape factors between about 2.5 and 4, depending on how the layer got there, which is inside the range the composite law spans. The model cannot say exactly where separation is, because separation is a zero of the wall friction and the friction here is never zero; what it does say is that a layer approaching separation is a layer whose profile is almost entirely wake — a free shear layer that happens to be resting on a wall. That is why a separating turbulent layer behaves so much like a mixing layer once it has left the wall, and why the large eddies, not the wall region, dominate what happens next. A free shear layer of that kind is unstable to the rolling-up that makes a sheet unable to stay a sheet, and the large spanwise rollers that appear behind a separation line are that instability, fed by a wake that has been growing all the way up the pressure rise.

How it was checked #

The composite layer at a wake strength of a thousand has a shape factor of 3.990 against the pure wake’s exact 4 — the wall law’s remaining share. Das’s fit gives 0.48 at $\beta = 0$, close to the flat plate’s measured wake, which is the fit’s own consistency with the previous essay. The quadrature is the previous essay’s, converged to a part in ten million in the shape factor. The exponent relation is algebra on the momentum integral and the definition of $\beta$, with the shape factor taken from the profile at each $\beta$.

The convention: β on the displacement thickness, m on the free stream #

Clauser’s parameter is $\beta = (\delta^*/\tau_w)\,dp/dx$, positive for a rising pressure; the free-stream exponent $m$ is negative for a decelerating stream. The momentum thickness is taken to grow in proportion to distance, which is exact for an equilibrium layer only in the limit of slowly varying friction; at a fixed Reynolds number of $10^4$ the friction’s drift with the layer’s growth is small and is ignored. The wall law’s constants are $\kappa = 0.41$ and $B = 5.0$.

What the picture cannot show #

The connection between $\beta$ and the wake strength is a fit to measurements, and fits disagree by tens of per cent at large $\beta$, where few equilibrium layers have been built; every number past $\beta \approx 10$ inherits that. The log law itself is assumed to survive in the wall region at every $\beta$. Measurements say it does at moderate gradients and is distorted close to separation, where the wall region thins and its own velocity scale is no longer the friction velocity alone. And no real diffuser or wing holds $\beta$ fixed: a layer whose $\beta$ is changing carries the wake of an earlier $\beta$ for a long distance — the outer region’s memory again — so an equilibrium layer is a reference case, and a real one is always catching up with it. Everything is two-dimensional as well. On a swept wing the same rising pressure turns the slow fluid near the wall towards the chord while the outer flow keeps its spanwise component, so the layer is skewed as it thickens — the crossflow that a swept wall makes along its isobars — and an equilibrium defined by one parameter in the plane of the flow is then an approximation to a layer whose profile also twists with height.

Who found it, and when #

Clauser’s equilibrium layers are from 1954 and his definition of $\beta$ from the same work; Coles’s wake from 1956 gave the outer profile a form to fit. Townsend and Mellor and Gibson in the 1960s worked out equilibrium layers theoretically with an eddy viscosity and found the same limiting behaviour as separation is approached. Stratford’s zero-friction layer of 1959 showed it could be built. Das’s fit of the wake strength to $\beta$ is one of several; the extremum of the free-stream exponent follows from the momentum integral and the shape factor whichever fit is used. Clauser’s own tunnel, with its flexible wall adjusted by hand until the outer profile stopped changing, remains the clearest demonstration that the equilibrium is a real state of a turbulent layer and not a convenience of the algebra.

Still open: a layer that is not in equilibrium #

Every layer here is matched to its gradient at every station. The next calculation releases one: a flat-plate layer meeting a sudden adverse gradient — a step in $\beta$ — with the wake strength relaxing towards its new equilibrium value over a distance set by the outer eddies’ turnover, and asks how many layer thicknesses the relaxation takes, how far the friction overshoots on the way, and whether a layer hit by a gradient steeper than the equilibrium limit separates before its wake has had time to grow — which would say how much of a diffuser’s stall margin is in the gradient’s shape and how much in its history.